Improved Bounds for the Union of Locally Fat Objects in the Plane
Boris S. Aronov, Mark de Berg, Esther E. Ezra, Micha Sharir · SIAM Journal on Computing · 2014
We show that, for any $\gamma > 0$, the combinatorial complexity of the union of $n$ locally $\gamma$-fat objects of constant complexity in the plane is $\frac{n}{\gamma^4} 2^{O(\log^*n)}$. For the special case of $\gamma$-fat triangles, the bound improves to $O(n \log^*{n} + \frac{n}{\gamma}\log^2{\frac{1}{\gamma}})$.